Solve 1/x^2+2(1/x^2)(1/y^2)+1/y^2

$$\frac{ 1 }{ { x }^{ 2 } } +2( \frac{ 1 }{ { x }^{ 2 } } )( \frac{ 1 }{ { y }^{ 2 } } )+ \frac{ 1 }{ { y }^{ 2 } }$$

Evaluate

$$\frac{x^{2}+y^{2}+2}{\left(xy\right)^{2}}$$

Short Solution Steps

Express 2\times \frac{1}{x^{2}} as a single fraction.

$$\frac{1}{x^{2}}+\frac{2}{x^{2}}\times \frac{1}{y^{2}}+\frac{1}{y^{2}}$$

Multiply \frac{2}{x^{2}} times \frac{1}{y^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{x^{2}}+\frac{2}{x^{2}y^{2}}+\frac{1}{y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and x^{2}y^{2} is x^{2}y^{2}. Multiply \frac{1}{x^{2}} times \frac{y^{2}}{y^{2}}.

$$\frac{y^{2}}{x^{2}y^{2}}+\frac{2}{x^{2}y^{2}}+\frac{1}{y^{2}}$$

Since \frac{y^{2}}{x^{2}y^{2}} and \frac{2}{x^{2}y^{2}} have the same denominator, add them by adding their numerators.

$$\frac{y^{2}+2}{x^{2}y^{2}}+\frac{1}{y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}y^{2} and y^{2} is x^{2}y^{2}. Multiply \frac{1}{y^{2}} times \frac{x^{2}}{x^{2}}.

$$\frac{y^{2}+2}{x^{2}y^{2}}+\frac{x^{2}}{x^{2}y^{2}}$$

Since \frac{y^{2}+2}{x^{2}y^{2}} and \frac{x^{2}}{x^{2}y^{2}} have the same denominator, add them by adding their numerators.

$$\frac{y^{2}+2+x^{2}}{x^{2}y^{2}}$$

Factor

$$\frac{x^{2}+y^{2}+2}{\left(xy\right)^{2}}$$